prove that τ is the topology on ℝ

May 2010
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0
Define τ⊂ P(ℝ) as follows:
τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .
prove that τ is a topology on ℝ .
 
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Drexel28

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Nov 2009
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Define τ⊂ P(ℝ) as follows:
τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .
prove that τ is the topology on ℝ .
Is this \(\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:(-\pi,\pi)\subseteq U\right\}\)? What's wrong? clearly \(\displaystyle \varnothing,\mathbb{R}\in T\) if \(\displaystyle (-\pi,\pi)\subseteq U_\alpha\) then \(\displaystyle (-\pi,\pi)\subseteq\bigcup_{\alpha\in\mathcal{A}}U_\alpha\) and similarly for the intersection. Or, is this \(\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:\{-\pi,\pi\}\subseteq U\right\}\)? It's the same.